What is the difference between Alif and Hamzah?

Question
What is the difference between Alif and Hamzah? Can you please explain the differences and their usage model?
Jazak Allah

Answer
The true alif is a saakin letter after a fathah , as in the word  :   Some list the letter alif as in the Arabic alphabet  as to make clear that it is always preceded by a fathah and is always saakin. 

The regular hamzah   is a letter that looks like the head of the letter  and  is pronounced from the deepest part of the throat.  It can be written over or below different letters as a holding place for the hamzah, not part of the letter.  One of the letters the hamzah can "sit" on is the alif, such as in: , or it can be written on a ya': , or a as in: .  It can be written on a "tooth" demonstrated in the word: , or have no particular place to rest:

Please also see: http://www.abouttajweed.com/q240601.htm
Wa iyyaakum